Kepler's Discovery: Musical Harmonics and Regular Polygons

This implies that every stationary motion must have an integer number of complete waves, or rather a frequency multiple of the fundamental frequency. These possible oscillations differ from each other only in relation to the number of wave crests that form them. The number of wave crests is an indicator of the frequency of the resulting sound: the greater the number of crests, the higher its frequency will be. Since virtually every integer number of wave crests is possible, so ideally every frequency exactly multiple of the fundamental is possible.

What we have highlighted, and which is crucial in this process, is the fact that the two ends of the string are fixed and therefore unable to move. If one end of the string could move freely, the string would not emit any sound, which instead happens when both ends are fixed. The deep reason for this phenomenon lies in the hidden musical properties of the circle.

It was, in fact, Kepler who realized that by ideally joining the two ends of the string to form a circle, it was possible to obtain every natural harmonic by inscribing the corresponding regular polygon inside the circle. The vertices of a triangle would indicate the nodes of oscillation of the third harmonic, the vertices of a square those of the fourth, those of a pentagon those of the fifth, and so on (Fig. 14).

Every harmonic could thus be considered as originating from the inscription of a regular polygon having a number of sides equal to the number of the harmonic that was intended to be produced. The harmonic succession thus appeared entirely resolved geometrically with the succession of regular polygons. Furthermore, since the natural diatonic scale with its seven notes arises from the intervals of this harmonic succession, and from these seven notes then departs all the study of musical modes and harmony, this equivalence between harmonic succession and polygons constitutes a key for interpreting the foundations of all musical harmony on a geometric basis (Fig. 13).

As a pure example, we show how, inspired by this correspondence between polygons and harmonics, it is possible to assign to certain chords or musical compositions equivalent constellations of regular polygons. Let us choose, for example, a fundamental note from which to consider the following harmonics. We choose in this case the note C. Since the note G represents the third harmonic of a fundamental C, we can in this case represent this note through the use of a triangle, where C can be represented with a circle with a point, being here the fundamental note. Thus, the C-G chord can be represented with a triangle inscribed in a circle with a point, while the G-C chord with a circumscribed triangle. In the same way, the addition of an E can be visualized with a 5-pointed star because E represents in this context the fifth harmonic (Fig. 15).

Mathematical Note on the Circle, Pontryagin's Dual and Fourier Series. We want to write some mathematical clarifications for those who have at least a university mathematical background. Clearly these can be skipped without precluding understanding of the article in its essence. The relationship between musical harmonics and the circumference is much deeper than what may appear at first impression.

Delving mathematically into the acoustic phenomenon of the previous paragraph, we can say that a musical note can be modeled through the use of a periodic function which, without loss of generality, we can assume to have period $2\pi$, i.e., $f(x+2\pi) = f(x+2\pi)$. From classical analysis we know that every periodic function is decomposable into the sum of its harmonics through its Fourier series, i.e., $$f(x) = \sum_{n=-\infty}^{\infty} c_n e^{inx}$$ where the $c_n$ are complex coefficients easily obtainable. This series is a decomposition into harmonics of multiple frequency.